The goal of diagnosis is to recover \(\mathbf {D}_\mathrm {mat}\)  and \(\mathbf {H}_\mathrm {TX}\)  simultaneously, and thus the failure mask \(\mathbf {M}_\mathrm {mat}\)  can be retrieved from \(\mathbf {M}_\mathrm {mat} = \mathbf {D}_\mathrm {mat} \oslash \mathbf {H}_\mathrm {TX} + \mathbf {1}_{H \times W}\) , in which non-unity entries indicate faulty reflecting elements.
However, the channel and the failure deviation are coupled since both of them are unknown.
To recover \(\mathbf {H}_\mathrm {TX}\)  and \(\mathbf {D}_\mathrm {mat}\)  simultaneously, we need to introduce a-prior information on them.
Note that \(\mathbf {D}_\mathrm {mat} = \mathbf {H}_\mathrm {TX} \circ (\mathbf {M}_\mathrm {mat} - \mathbf {1}_{H \times W})\)  is a sparse matrix due to the failure sparsity that most entries in \(\mathbf {M}_\mathrm {mat}\)  are unity.
According to the channel assumption in Sec. REF  (sparse mmWave channel), \(\mathbf {H}_\mathrm {TX}\)  is a low-rank matrix since \(\mathrm {rank}(\mathbf {H}_\mathrm {TX}) \le L_\mathrm {TX} \ll \min (H,W)\) .
Therefore, \(\mathbf {X}=\mathbf {D}_\mathrm {mat} + \mathbf {H}_\mathrm {TX}\)  is the sum of a sparse and a low-rank matrix, which allows \(\mathbf {H}_\mathrm {TX}\)  and \(\mathbf {D}_\mathrm {mat}\)  to be recovered simultaneously from compressed measurements thanks to the rank-sparsity incoherence [1]}, [2]}.
